## Abstract

A novel subwavelength surface plasmon polaritons optical filter based on an incompletely directional coupler is proposed and numerically simulated by using the finite difference time domain method with perfectly matched layer absorbing boundary condition. An analytical solution for the resonant condition of the structure is derived by means of the cavity theory. Both analytical and simulative results reveal that the resonant wavelengths are proportional to the length of the slit segment, inversely proportional to the antinode number of a standing wave in the segment, and are related to the slit width and the gap between the two slits. The analytical solution being consistent with the numerical simulation verifies the feasibility of the concept of the new filter structure.

© 2009 OSA

## 1. Introduction

Surface plasmons (SPs), which are trapped propagating on the surfaces of metals due to their interaction between the free electrons in metals and photons, can confine and propagate the electromagnetic energy in a subwavelength limit [1,2]. At present, more and more researchers pay their attentions on the investigation of surface plasmon polaritons (SPPs). Thus, a lot of articles about SPPs are emerged, from which numerous devices based on SPPs is proposed, such as splitters [3], U-shaped waveguides [4], Y-shaped combiners [5], directional couplers [6,7], and many of those waveguides have been theoretically or experimentally studied. In order to realize wavelength selecting characteristics, Bragg-type structures based on SPPs have been proposed. They include the metal heterostructures constructed with several periodic slots placed vertically along an Metal-Insulator-Metal (MIM) waveguide [8,9], the Bragg grating fabricated by periodically modulating the thickness of thin metal stripes embedded in an insulator [10], and the periodic structure formed by changing alternately two kinds of the insulators with the same width [11] or different widths [12,13]. And a high-order plasmonic Bragg reflector with periodic modulation of the core index of the insulator [14] as well as a structure with periodic variation of the width of the insulator in an MIM waveguide [15] has been proposed. Most of the structures mentioned above, however, have period numbers of N>9 with the total lengths over 4 µm and beyond the subwavelength scale, which result in a relatively high insertion loss of several decibels. Recently, a tooth-shaped plasmonic waveguide filter with nanometer size [16] is proposed. It is of an ultracompact size with a few hundreds of nanometers in length and then low insertion loss. Moreover, it could reduce the difficulties of fabrication, compared with previous grating-like heterostructures with a few micrometers in length. In this paper, another novel optical wavelength filter based on a MIM coupler is proposed for the first time. The finite difference time domain (FDTD) method in a perfectly matched layer (PML) absorbing boundary condition is employed to simulate and study its new properties. This waveguide device could be highly integrated for its transverse size being subwavelength-scale.

## 2. Device structure and theoretical analysis

The optical filter structure is simply composed with a MIM directional-coupling slit-waveguide where one of the slits is only a short segment (shown in Fig. 1
). The slits’ medium is assumed to be air whose refractive index is set to be 1. The circumambience of the slits is covered with metal Ag. The frequency-dependent complex relative permittivity of silver is characterized by the Drude model ${\epsilon}_{m}\left(\omega \right)={\epsilon}_{\infty}-{\omega}_{P}/\omega \left(\omega +i\gamma \right)$.Here${\omega}_{P}=1.38\times {10}^{16}$ Hz is the bulk plasma frequency, which represents the natural frequency of the oscillations of free conduction electrons; $\gamma =2.37\times {10}^{13}$Hz is the damping frequency of the oscillations, *ω* is the angular frequency of the incident electromagnetic radiation, and ${\epsilon}_{\infty}$stands for the dielectric constant at infinite angular frequency with a value of 3.7 [15]. When incident optical wave from point O transmits along slit waveguide 1, parts of the wave is coupled into slit segment 2, where the forward and backward waves are almost completely reflected in the air-Ag interfaces at the two ends of the short waveguide segment. Parts of the waves are coupled back into slit waveguide 1. Therefore, the structure is like a resonance cavity, and standing-waves can be formed with some appropriate conditions in the short waveguide segment. Finally, the incident optical wave is converted into two parts, the reflected wave and the transmitted wave, by the structure. Defining $\Delta \varphi $ to be the phase delay per round-trip in the ‘cavity’, one has $\Delta \varphi =k(\omega )\u20222L+2{\varphi}_{{}_{ref}}$. Where, $k(\omega )$ is the angular wavenumber of the wave in the segment at frequency *ω*, and *L* is the length of the slit-waveguide segment. ${\varphi}_{ref}$is the phase shift of a beam reflected on air-metal interface at each end of the segment. Stable standing waves can only build up constructively within the “cavity” when the following resonant condition is satisfied, based on the principle of a resonant cavity:

*m*is the number of antinodes of the standing SPPs wave. Considering$k(\omega )=\frac{2\pi}{{\lambda}_{}}{n}_{eff}$, one can get the resonant wavelengths to bewhere, ${n}_{eff}$ is the effective index of the slit-waveguide segment in coupling with another slit waveguide, λ and λ

_{m}are vacuum wavelengths of the wave. From Eq. (2), it can be concluded as following: 1) The ratio of ${\lambda}_{1}$:${\lambda}_{2}$:${\lambda}_{3}$is equal to 1:2:3, as the term of ${\varphi}_{ref}/\pi $ is very small. 2) The resonant wavelengths are all proportional to the length

*L*of the “resonant cavity” segment, but with different slope factor of 1

*/m*. 3) The wavelengths of ${\lambda}_{m}$are all proportional to the effective index ${n}_{eff}$. Because ${n}_{eff}$ is correlated inversely with the width of a MIM waveguide [8,16] (that is, ${n}_{eff}$decreases with the increase of the width), one can expect that resonant wavelengths are also correlated inversely with the width

*W*. 4) As a directional coupler structure, the effective index ${n}_{eff}$of the waveguide segment also depends on the coupling strength of the MIM coupler. Therefore there is a certain relation between the effective index ${n}_{eff}$and the gap between the two coupling waveguides, and the resonant wavelengths are dependent on the gap in some way.

## 3. Simulation experiment and results

To verify the above filtering concept and its theoretical analysis, the FDTD method with perfectly matched layer absorbing boundary condition is used to simulate the MIM coupling structure. In the following simulations, the grid sizes in the *x* and the *z* directions are chosen to be 5nm and 5nm. The length of waveguide 2 is assumed to be *L*. The fundamental TM mode of the plasmonic waveguide is excited by dipole source at location A. Time monitors, set wider than the width of the long slit waveguide in points A and B, are used to detect the reflected and transmitted powers of *P _{ref}* and

*P*at the locations. The vacuum wavelength of the incident light wave is scanned to find spectrum responses of the structures. The transmittance and reflectivity of the structure are defined to be R

_{tr}*= P*, and T

_{ref}/ P_{in}*= P*, respectively.

_{tr}/ P_{in}At first, we keep the widths of the two waveguides having the same value of 50nm. The length *L* of the waveguide segment is 0.5µm. The gap between the two waveguides is set to be 20 nm. Figure 2
shows the spectra of the transmission and the reflection of the structure, with the wavelength-selective characteristic of a typical filter. The minimum transmittance and the maximum reflectivity occur at the wavelengths of 0.78µm and 1.57µm, respectively. For a non-coupling slit waveguide with a width of 50 nm, its effective index *n _{eff}* is calculated to be about 1.5, based on the

*n*equations in References [10,17]. Thus, the wavelengths of the transmittance/reflectivity extremas of the coupler-type structure are respectively given to be 0.75μm for

_{eff}*m =*2 and 1.5μm for

*m =*1 with the length of

*L =*500 nm of the segment and a provided phase-shift of ${\varphi}_{ref}=0$, according to Eq. (2).

Figures 3 (a) and 3(b) show the spectra and the peak positions of the reflectivity of the structure at different gaps, and with the fixed slit width of 50 nm and the segment length of 500 nm. It reveals that the wavelengths of the peaks shift toward short wavelength slowly with the increasing of the gap, and the bandwidths of peaks become a bit narrow at the same time. However, with the increase of the gap, the maximum transmittances are reduced owing to the fact that the loss of a MIM coupler will increase with its gap [7]. The phenomenon of gap-dependent peak wavelengths can be attributed to the weak dependence of the effective index ${n}_{eff}$ on the gap, as mentioned in the discussion of Eq. (2). Our simulation also reveals that the filtering spectrum of the structure will be distorted badly if the gap is smaller than 18nm. It means very large coupling strength will weaken the “cavity” effect, due to large amount of the energy coupling out of the segment.

Figures 4 (a)
and 4 (b) show the dependences of the peak wavelengths of the reflective spectrum (or the wavelengths of the minima of the transmitted spectrums) on length *L* and width *W*, respectively. As is revealed in Eq. (2), the wavelengths of the two reflection peaks increase linearly with the increasing of the segment length of *L*, and correlate inversely with the width of *W*. However, the change rate of the first peak wavelength is about twice of that of the second peak wavelength. Obviously, one can realize the function of selecting any wavelength using the way of changing the parameters of the device, such as the waveguide width, the length of the slit segment, or the gap between the slit waveguides.

The quality factor of $Q=2\pi \nu \frac{\text{E}}{P}=\frac{\nu}{\Delta \nu}$is an important parameter for a cavity. Here, E is the stored energy in the cavity, and *P* is the power dissipated. $\nu =c/\lambda $is a resonant frequency of the cavity, *λ* is the resonant wavelength, and *c* is the speed of light. Δν is the bandwidth of a resonant peak of the cavity. Figures 5(a) and (b)
show the dependences of the quality factor on the gap and the width of the segment, respectively. It can be seen that the quality factor increases with the increasing of the gap and the width of the segment. The increment of *Q* becomes small for large values of the gap and the width of the segment. The reason is that the coupling strength of the MIM coupler decreases with increasing of the gap, so that the power dissipating from the “cavity” segment decreases. The increasing of the segment width will enhance the asymmetry of the coupler, and thus the coupling strength decreases.

## 4. Summary

In conclusion, a novel SPPs wavelength filter based on an incomplete coupler structure is proposed for the first time. The simulation demonstrates that the device has typical characteristics of a wavelength filter. The simple structure with subwavelength size would be helpful to reduce fabrication difficulties compared with previous grating-like heterostructures with a few micrometers in length. The new structure may have applications to ultrahigh nanoscale integrated photonic circuits on flat metallic surfaces.

## Acknowledgment

The authors acknowledge the financial support from the Natural Science Foundation of Guangdong Province, China (Grant No. 07117866).

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